Let \(p, L = {Q}(\beta ), g(x), \lambda \)  be as in Theorem REF . Let \(\lambda = -\frac{\ell }{e}\)  with \(\gcd (\ell , e) = 1\)  and \(g(x)\)  is not \(p\) -regular, i.e., the residual polynomial \(T(g)(Y) \in \mbox{$\mathbb {F}$}_p[Y]\)  of \(g(x)\)  has repeated roots. In this situation, as in [1]}, we define the Newton polygon of second order. Let \(T(g)(Y)\)  is a power of monic irreducibile polynomial \(\psi (Y)\)  over \(\mbox{$\mathbb {F}$}_p\) , i.e., \(T(g)(Y) = \psi (Y)^{s}\)  in \(\mbox{$\mathbb {F}$}_p[Y]\) , where \(s \ge 2\) . In this case, we construct a key polynomial \(\phi (x)\)  attached with the slope \(\lambda \)  such that the following hold:
